An algebraic approach to problems with polynomial Hamiltonians on Euclidean spaces

TL;DR

This paper provides explicit algebraic formulas for matrix elements of polynomial Hamiltonians on Euclidean spaces, enabling analytical solutions and optimal basis selection for quantum problems involving SU(1,1) and SO(N) symmetries.

Contribution

It introduces explicit algebraic expressions for matrix elements of radial and orbital observables in multi-dimensional spaces, facilitating analytical solutions of polynomial Hamiltonians.

Findings

Explicit formulas for SU(1,1) radial matrix elements.

Explicit formulas for SO(N)-reduced matrix elements.

Method for optimal basis selection to improve convergence.

Abstract

Explicit expressions are given for the actions and radial matrix elements of basic radial observables on multi-dimensional spaces in a continuous sequence of orthonormal bases for unitary SU(1,1) irreps. Explicit expressions are also given for SO(N)-reduced matrix elements of basic orbital observables. These developments make it possible to determine the matrix elements of polynomial and a other Hamiltonians analytically, to within SO(N) Clebsch-Gordan coefficients, and to select an optimal basis for a particular problem such that the expansion of eigenfunctions is most rapidly convergent.